Physics-informed cluster analysis and a priori efficiency criterion for the construction of local reduced-order bases

TL;DR

This paper introduces a physics-informed clustering method for constructing local reduced-order bases, demonstrating improved performance over traditional methods in nonlinear physics problems, and provides an a priori efficiency criterion for ROM-net evaluation.

Contribution

It presents a novel clustering approach using angle-based dissimilarity and an a priori criterion for assessing ROM-net relevance, enhancing local reduced-order basis construction.

Findings

Clustering with angle-based dissimilarity reduces intra-cluster Kolmogorov N-width.

The a priori efficiency criterion effectively evaluates ROM-net relevance.

The proposed method outperforms classic strategies in five physics problems.

Abstract

Nonlinear model order reduction has opened the door to parameter optimization and uncertainty quantification in complex physics problems governed by nonlinear equations. In particular, the computational cost of solving these equations can be reduced by means of local reduced-order bases. This article examines the benefits of a physics-informed cluster analysis for the construction of cluster-specific reduced-order bases. We illustrate that the choice of the dissimilarity measure for clustering is fundamental and highly affects the performances of the local reduced-order bases. It is shown that clustering with an angle-based dissimilarity on simulation data efficiently decreases the intra-cluster Kolmogorov $N$-width. Additionally, an a priori efficiency criterion is introduced to assess the relevance of a ROM-net, a methodology for the reduction of nonlinear physics problems introduced in our previous work in [T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck, Model order reduction assisted by deep neural networks (ROM-net), Advanced Modeling and Simulation in Engineering Sciences 7 (16), 2020]. This criterion also provides engineers with a very practical method for ROM-nets' hyperparameters calibration under constrained computational costs for the training phase. On five different physics problems, our physics-informed clustering strategy significantly outperforms classic strategies for the construction of local reduced-order bases in terms of projection errors.